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Revised June 2013 Use polynomial identities to solve problems Rewrite rational expressions HSA-APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples HSA-APR.D.6 Rewrite simple rational expressions in different forms; writea(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods graph equations on coordinate axes with labels and scales Create equations that describe numbers or HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; relationships Arithmetic with Polynomials and Rational Expressions HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions HSA-APR.D.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions HSA-APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial Understand the relationship between zeros and factors of polynomials Write expressions in equivalent forms to solve problems Seeing Structure in Expressions HSA-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x) HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression: a) Factor a quadratic expression to reveal the zeros of the function it defines, b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines, c) Use the properties of exponents to transform expressions for exponential functions HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2) The Complex Number System HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context: a) Interpret parts of Interpret the an expression, such as terms, factors, and coefficients, b) Interpret complicated expressions by viewing structure of one or more of their parts as a single entity expressions polynomials Use complex numbers in HSN-CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as polynomial (x + 2i)(x – 2i) identities and equations HSN-CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic HSN-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions HSN-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers HSN-CN.A.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers Quantities HSN-CN.A.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling The Real Number System Perform arithmetic operations with complex numbers Reason quantitatively and use units to solve problems HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays HSN-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents Extend the properties of exponents to rational exponents HSN-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents CCSS Math Quick Guide Algebra II Creating Equations Revised June 2013 HSF-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context HSF-LE.A.4 For exponential models, express as a logarithm the solution toabct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology HSF-LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table) Interpret expressions for functions in terms of the situation they model Construct and compare linear, quadratic, and exponential models and solve problems Building Functions HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions: a) Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals, b) Recognize situations in which one quantity changes at a constant rate per unit interval relative to another, c) Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another HSF-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x),f(kx), and f(x + k) for specific values Build new functions of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of from existing the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic functions expressions for them HSF-BF.B.4 Find inverse functions: a) Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse, b) Verify by composition that one function is the inverse of another, c) Read values of an inverse function from a graph or a table, given that the function has an inverse, d) Produce an invertible function from a noninvertible function by restricting the domain Build a function that models a relationship between two quantities Interpreting Functions HSF-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms HSF-BF.A.1 Write a function that describes a relationship between two quantities: a) Determine an explicit expression, a recursive process, or steps for calculation from a context, b) Combine standard function types using arithmetic operations, c) Compose functions HSF-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions HSF-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function: a) Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context, b) Use the properties of exponents to interpret expressions for exponential functions. HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using Analyze functions technology for more complicated cases: a) Graph linear and quadratic functions and show intercepts, maxima, and minima, using different b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, c) representations Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior, d) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior, e) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes Interpret functions that arise in applications in terms of the context Understand the concept of a function and use function notation inequalities graphically Reasoning with Equations and Inequalities HSF-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity HSF-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers HSF-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context HSF-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of fcorresponding to the input x. The graph of f is the graph of the equation y = f(x) HSA-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions HSA-REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater) HSA-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the Represent and solve coordinate plane, often forming a curve (which could be a line) equations and HSA-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable HSA-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically HSA-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables HSA-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions HSA-REI.B.4 Solve quadratic equations in one variable: a) Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form, b) Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b Solve system of equations Solve equations and inequalities in one variable HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters HSA-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise Understand solving equations as a process of reasoning and explain the reasoning HSA-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method CCSS Math Quick Guide Algebra II Linear, Quadratic, and Exponential Models Model periodic phenomena with trigonometric functions Extend the domain of trigonometric functions using the unit circle Revised June 2013 HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots) HSS-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers) HSS-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems Use rules of probability to compute HSS-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the probabilities of answer in terms of the model compound events in a uniform HSS-CP.B.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) probability model = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model HSS-CP.B.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model HSS-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations HSS-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities Understand independence and conditional probability and use them to interpret data Making Inferences and Justifying Conclusions HSS-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not” HSS-CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent HSS-CP.A.3 Understand the conditional probability of A given B as P(A andB)/P(B), and interpret independence of A and B as saying that the conditional probability of A givenB is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B Interpreting Categorical and Quantitative Data HSS-IC.B.6 Evaluate reports based on data Understand and evaluate random processes HSS-IC.A.2 Decide if a specified model is consistent with results from a given data-generating underlying process, e.g., using simulation statistical experiments HSS-IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and Make inferences observational studies; explain how randomization relates to each and justify HSS-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a conclusions from sample surveys, margin of error through the use of simulation models for random sampling experiments and HSS-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to observational decide if differences between parameters are significant studies HSS-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related: a) Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models, b) Informally assess the fit of a function by plotting and analyzing residuals, c) Fit a linear function for a scatter plot that suggests a linear association HSS-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population HSS-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data HSS-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve Expressing Geometric Properties w/ Equations Summarize, represent, and interpret data on two categorical and quantitative variables Summarize, represent, and interpret data on a single count or measurement variable HSG-GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix Translate between the geometric description and the equation for a conic sections identities Trigonometric Functions HSG-GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation HSF-TF.C.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems HSF-TF.C.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or Prove and apply tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle trigonometric HSF-TF.B.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context HSF-TF.B.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed HSF-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline HSF-TF.A.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions HSF-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle HSF-TF.A.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real numbe HSF-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle CCSS Math Quick Guide Algebra II Conditional Probability and the Rules of Probability CCSS Math Quick Guide Algebra II Standards of Mathematical Practice These standards should be integrated throughout teaching and learning of the content standards of the Common Core State Standards. 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically. 7. Look for and make use of structure. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 6. Attend to precision. 8. Look for and express regularity in repeated reasoning. Cluster Key Major Cluster (at least 75% of instructional time) Supporting Cluster Additional Cluster Revised June 2013